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Lorina Varvaruca\\
University of Bath\\
mapel@maths.bath.ac.uk\\

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{\bf On the Quasi-convexity of a functional in Nonlinear Elasticity}\\


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\noindent {\bf Abstract}: Let $\Omega\subset\mathbb{R}^3$ denote the
reference configuration of a nonlinear elastic material, and consider
deformations$\mbox{\boldmath $u$}:\Omega\to \mathbb{R}^3$. We
determine a sufficient condition for the stored energy density $W({\bf
F})=|{\bf F}|^p+h(\det{\bf F})$, where $h$ is a smooth and convex
function, and $2< p<3$, to be $W^{1,p}$-quasiconvex at $\lambda {\bf
I}$ on the restricted class of deformations \mbox{\boldmath $u$} that satisfy
condition (INV), $\det\nabla \mbox{\boldmath $u$}>0$ a.e., and which
open a single hole anywhere in the material. The condition takes the
form $\lambda^{3-p}h'(\lambda^3)\leq \Upsilon_p$, where $\Upsilon_p>0$ is
explicitly determined, and is in a certain sense optimal for the class
of energy densities under consideration. The proof makes use of
isoperimetric inequalities and of estimates on radial $p$-harmonic
functions.
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